ProfinitePlot

The graph of $f: \hat{\mathbb{Z}} \to \hat{\mathbb{Z}}$ in the variable $x$, at precision $k$ (default 5) — Lenstra's picture of a profinite function, drawn as an ArrayPlot.

ProfinitePlot(f, x, k)residue classes mod $k!$ laid out by their factorial digits

ProfinitePlot(f, x)default precision $k = 5$

Typed signatures

Type	Package	Applies
(expression, symbol, integer?) -> expression<ArrayPlot>	adeles	always

Domain: Adèles and idèles

Details

• Each cell is a pair of residue classes mod $k!$; a cell is filled when $f$ maps its column class into its row class, evaluating $f$ on ProfiniteNumber(a, k!) for every $a$.
• $k$ is capped at 6 ($6! = 720$ cells a side) to keep the plot tractable.
• The result is an ArrayPlot of 0/1 cells, as in Hertogh's ProfiniteGraph.

Examples

Basic #

the identity function: an anti-diagonal of filled cells

live ↗

In$\mathrm{ProfinitePlot}(x, x, 3)$Out$\mathrm{ArrayPlot}(\bigl\lbrack\bigl\lbrack0, 0, 0, 0, 0, 1\bigr\rbrack, \bigl\lbrack0, 0, 0, 0, 1, 0\bigr\rbrack, \bigl\lbrack0, 0, 0, 1, 0, 0\bigr\rbrack, \bigl\lbrack0, 0, 1, 0, 0, 0\bigr\rbrack, \bigl\lbrack0, 1, 0, 0, 0, 0\bigr\rbrack, \bigl\lbrack1, 0, 0, 0, 0, 0\bigr\rbrack\bigr\rbrack)$

Possible issues #

$k$ is capped at 6 to keep the plot tractable; asking for more leaves the call standing

live ↗

In$\mathrm{ProfinitePlot}(x, x, 9)$Out$\mathrm{ProfinitePlot}(x, x, 9)$

About

Examples	2

See also: ProfiniteNumber, ArrayPlot